Backprojection

In order to ensure perpendicular beam incidence on the cylindrical specimen, the circular B-scan profiles were acquired by high frequency (narrow beam) transducers in a synthetic circular aperture array. From these profiles two-dimensional reflection tomograms were reconstructed using a filtered backprojection technique. Straight line propagation was assumed. Several artificial discontinuity types in a cylindrical Plexiglas (Perspex) specimen were compared with similar artificial discontinuities in a cylindrical A/Si-alloy [2]. Furthermore, examples of real discontinuities (an inclusion and a feed head) in the cylindrical AlSi-alloy are presented. 

Peyrin F. The generalized backprojection theorem for cone-beam reconstruction., IEEE Trans. Nucl. Sci., V. NS-32, 1985, p.1512-1519. 

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

Keywords radar radar imaging tomography high resolution synthetic aperture radar interferometry polarimetry Radon transform projection slice theorem backprojection. 

The Filtered Backprojection (FBP) method may be used to process by reconstructing the original image from its projections in two steps Filtering and Backprojection. 

The expression x Vv — Xypu is the Jacobian J(pj ) of the backprojection mapping pj along the z-axis, from the plane (11) to Tl at point P. By definition of the Jacobian, m being the measure of the ball B, of radius r, we have ... 

In two dimensions, Euclidean balls are disks whose measures are areas, so that J Pz ) is actually the backprojection ratio 7Z that can be computed easily from the normal... 

The first point has not been clearly recognized or appreciated in the early days of the method, so terms like radon transformation aad filtered backprojection were introduced [Herl, Manl], In practical realizations of image reconstruction from projections, however, numerical frers must, indeed, be used [Herl]. 

Use of the expression backprojection instead of reconstruction from projections is historical. Given a sufficient number of projections of an object acquired at different angles, the shape of the object can indeed be reconstructed with recognizable features, if the projections are just smnmed over the image plane in the directions over which... 

The backprojection approach uses straightforward addition of the projections acquired at different angles

discrete variable for simplicity, the backprojection image Mq is obtained by integration over [Pg.202]

A linear filter performs a convolution of the input function with the Fourier transform of the filter transfer function. According to the convolution theorem (cf. Section 4.2.3) application of a filter in one domain corresponds to multiplication of the Fourier transform of the function to be filtered with the filter-transfer function. To filter a backprojection image, eqn (6.1.3) is Fourier transformed,... 

Because cylinder coordinates are used, the scaling factor lk = k appears in the integral. Apart from this and the prefactor (2n), eqn (6.1.5) and the backprojection frnmula (6.1.3) are identical. 

Multiplication of the FID p k, p) by Ikl prior to Fourier transformation for use by the backprojection method (6.1.3) can be interpreted in terms of filtering the projection P r, o) by a filter the transfer function of which is given by f k) = k. For this reason, calculation of the image by proper transformation of the raw data from cylindrical to... 

Different approaches can be taken to obtain radial images. Radial field gradients can be applied by the use of dedicated hardware [Hakl, Leel, Lee2]. Alternatively, a 2D image can be reconstructed from one projection by the backprojection technique, and a radial cross-section can be taken through it. The most direct way to access the radial image from a projection consists in computing the inverse Hankel transformation (cf. Section 4.4.2) of the FID measured in Cartesian k space (cf. Fig. 4.4.1) [Majl]. But in practice, the equivalent route via Fourier transformation of the FID and subsequent inverse Abel transformation (cf. Section 4.4.3) is preferred because established phase and baseline correction routines can be used in the calculation of the projection as an intermediate result. 

Typical flip angles are in the order of 15° for the slice-selective excitation pulses. The intensity of the FID after such a pulse corresponds to 25% (sin 15°) of the intensity after a 90° pulse. However, more than 96% (cos 15°) of the longitudinal magnetization are preserved, enabling fast repetition rates. The method can be employed in combination with the backprojection imaging scheme and with spin-warp imaging by phase encoding of spatial information. 

For each image pixel (x,y) at each projection angle o, r is calculated by Eq. (4.1). The measured counts in the projection sinogram corresponding to the calculated r are added to the (x,y) pixel in the reconstruction matrix. This is repeated for all projection angles. Thus, the backprojected image pixel... 

Figure 4.1. Principle of backprojection in image reconstruction, (a) An object with two hot spots (solid spheres) is viewed at three projection angles (at 120° angle) and the acquired data are backprojected for image reconstruction (b) When many views are obtained, the reconstructed image represents the activity distribution with hot spots, but the activity is blurred around the spots (c) Blurring effect described by 1 /r function, where r is the distance away from the central point. (Reprinted with the permission of The Cleveland Clinic Center for Medical Art Photography 2009. All Rights Reserved)...

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